Minimum Mutual Information and Non-Gaussianity through the Maximum Entropy Method: Estimation from Finite Samples

Pires C., Perdig„o R.A.P.
Entropy 2013, 15(3), 721-752, doi:10.3390/e15030721

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The Minimum Mutual Information (MinMI) Principle provides the least committed, maximum-joint-entropy (ME) inferential law that is compatible with prescribed marginal distributions and empirical cross constraints. Here, we estimate MI bounds (the MinMI values) generated by constraining sets Tcr comprehended by mcr linear and/or nonlinear joint expectations, computed from samples of N iid outcomes. Marginals (and their entropy) are imposed by single morphisms of the original random variables. N-asymptotic formulas are given both for the distribution of cross expectationís estimation errors, the MinMI estimation bias, its variance and distribution. A growing Tcr leads to an increasing MinMI, converging eventually to the total MI. Under N-sized samples, the MinMI increment relative to two encapsulated sets Tcr1 < Tcr2 (with numbers of constraints mcr1 < mcr2) is the test-difference deltaH = Hmax1-Hmax2 >= 0 between the two respective estimated MEs. Asymptotically, deltaH follows a Chi-Squared distribution (1/2N)X^2(mcr1-mcr2) whose upper quantiles determine if constraints in Tcr2/Tcr1 explain significant extra MI. As an example, we have set marginals to being normally distributed (Gaussian) and have built a sequence of MI bounds, associated to successive non-linear correlations due to joint non-Gaussianity. Noting that in real-world situations available sample sizes can be rather low, the relationship between MinMI bias, probability density over-fitting and outliers is put in evidence for under-sampled data.